Bayesian Structural VAR models: an extended approach. February 28, PRELIMINARY DO NOT CIRCULATE (most recent version available here)

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1 Bayesian Structural VAR models: an extended approach Martin Bruns Michele Piffer February 8, 18 - PRELIMINARY DO NOT CIRCULATE (most recent version available here) Abstract We provide an approach to derive the posterior distribution of SVAR models directly on the structural parameters. Our framework allows for restrictions not only on the contemporaneous relations among variables (as already in Sims and Zha, 1998 and Baumeister and Hamilton, 15), but also on impact impulse responses. In so doing, we enrich the tool available for the researchers who aim to set identify structural VAR models. Applying the methodology to simulated data on the New Keynesian model, we find that our approach recovers the true responses more tightly than the popular indirect orthogonal reduced form approach. We then apply our procedure to the identification of fiscal shocks. JEL classification:. Keywords:. We are thankful to Lutz Kilian, Helmut Lütkepohl and Haroon Mumtaz for helpful comments and suggestions. Many thanks to Taiki Yakamura for excellent research assistance. Michele Piffer thanks for the financial support received from the European Union s Horizon research and innovation program, Marie Sklodowska-Curie grant agreement number Freie Universität Berlin and German Institute for Economic Research (DIW Berlin). Queen Mary, University of London. Corresponding author m.b.piffer@gmail.com

2 1 Introduction Structural Vector Autoregressive models (SVARs) are typically specified in one of the following three forms, Ay t = Cw t + ɛ t, ɛ t N(, D), (1) y t = Πw t + Bɛ t, ɛ t N(, D), () Ay t = Cw t + Bɛ t, ɛ t N(, D), (3) with y t a vector of variables, w t = (1, y t 1,.., y t p) a vector of lagged variables, ɛ t a vector of structural shocks, and D a diagonal matrix. As discussed in Lütkepohl (5), Amisano and Giannini (1) and Caldara and Kamps (17), whether the researcher uses the A form, the B form, or the AB form (equations (1), () and (3), respectively) depends on the identifying restrictions. For example, since the A matrix in equation (1) captures the contemporaneous relation among variables, restrictions on such relations are more naturally imposed in the A form. By contrast, if the researcher aims to impose restrictions on the overall effect of the shocks, it is more natural to specify the model in the B form. The distinction is important because, as an example, zero restrictions on a matrix do not always imply zeros on its inverse. 1 A popular approach to study SVARs in a Bayesian framework consists of specifying prior distributions on the reduced form of the model, y t = Πw t + u t, u t N(, Σ). (4) 1 A notable special case is the recursive identification, for which a triangular matrix A in the A form implies a triangular matrix B in the B form. As an example of the opposite case, consider the A model a 11 a 1 by Sims and Zha (6), which imposes the non recursive zero restrictions A = a 1 a. a 31 b 11 b 1 a 33 It holds that, in general, the corresponding inverse matrix A 1 = B = b 1 b will only b 31 b 3 b 33 display two zero entries 1

3 Then, one uses draws of orthogonal matrices to map reduced form draws into the structural form of interest, be it the A form, the B form or the AB form (Arias et al., forthcoming). Due to its flexibility and tractability, the orthogonal reduced form approach has been extensively used in applied work (Uhlig, 5, Dedola and Neri, 7). However, as outlined by Baumeister and Hamilton (15), it suffers from an important drawback, namely that the prior on the structural parameters indirectly implied by the priors on the reduced form imposes restrictions which potentially go beyond the intention of the researcher. Such restrictions arise as a byproduct of the fact that priors are not expressed directly on the parameters of interest. For example, one might intentionally impose that a stuctural parameter is positive, but fails to control for the fact that the scale of the parameter implicit in the prior is excessively high and inconsistent with the researcher s prior information. Building on Sims and Zha (1998), Baumeister and Hamilton (15) derive a posterior sampler directly for the parameters in the A form from equation (1), and defend a more direct approach that specifies priors on the structural parameters of interest. This paper derives the posterior distribution for SVAR models in the AB form from equation (3), which nests the A form and the B form as special cases. In doing so, the paper aims to contribute to the literature by providing flexibility on the prior information for the structural parameters in the B form or in the AB form. This is relevant for at least three reasons. First, researchers frequently impose restrictions using the B form rather than the A form. Indeed, sampling the papers that use SVAR models published on top five journals in the last years, we found that as many as 71% use the B specification of the model. Second, the literature has expressed an interest in identifying restrictions on impulse responses at different possible horizons. Since the relation between the impulse responses at horizon h > and the elements Out of the 4 issues that we checked since 1998, we found that around 1% employs Structural Vector autoregressive models. Of this 1%, approximately 1% specifies the model in the A form, 7% specifies the model in the B form, and 5% specifies the model in the AB form. The details list is available here [ADD HYPERLINK].

4 in A, B, C is nonlinear, imposing explicit priors on impulse responses raises nontrivial technical challenges (see Plagborg-Møller et al., 16 for an analysis in a general framework, and Kociecki, 1 for the special case of a recursive structure). Since the B matrix in the B form from equation () captures the impulse responses on impact (i.e. for h = ), the analysis in this paper offers a compromise that allows for restrictions on impact impulse responses without facing the technical challenges associated with the moving average representation of the model. Third, since extending the Bayesian linear SVAR to a nonlinear framework is less challenging when departing from the direct structural approach rather than from the orthogonal reduced form approach, extending the set of model specifications that can be used enriches the tools available for nonlinear analysis, as we show in Bruns and Piffer (17). An important advantage of the orthogonal reduced form approach is that posterior simulation is not computationally demanding, because there exist priors such that marginal or conditional posteriors take common forms and can be drawn from directly. By contrast, when prior distributions are specified on the structural parameters, posterior simulation is technically more challenging as soon as the researcher moves away from the restrictive case of a recursive model (Kociecki et al., 1 provide an analysis of the latter). In the set identified models considered in the paper, the posterior is potentially irregular and has multiple peaks and ridges, a problem that increase in the dimensionality of the model. We address this problem by using the dynamic striated Metropolis-Hastings sampler by Waggoner et al. (16). Building on the literature on sequential Monte Carlo algorithms, the sampler gradually transforms the prior into the posterior using the draws at each stage to select the starting point of the Metropolis-Hastings for the next stage. Compared to the application in Waggoner et al. (16), we require the sampler only for a small subset of the parameters of the model, a feature that makes our approach feasible also for medium sized models. After deriving and discussing the posterior distribution for the general model in the AB form, we develop two applications. We first employ a simulation exercise to 3

5 compare our approach with the orthogonal reduced form approach. Since we use the New Keynesian model by An and Schorfheide (7) and since the shocks in such models are frequently identified on real data using the B form, we employ the B specification of the SVAR for such application, under the normalization of D = I. Log scores reveal that our approach delivers impulse response bands that cover the true impulse response more frequently and more tightly than the orthogonal reduced form approach. We then move to real data and employ the AB form of the model to replicate the analysis of fiscal shocks by Blanchard and Perotti (). We use the approach derived in our paper to relax selected identifying restrictions that Blanchard and Perotti () use to achieve exact identification. The paper is structured as follows. Section derives the model and the posterior distribution. Section 3 discusses the application to speudo data simulated from the New Keyesian model. Section 4 shows the application to fiscal shocks. The model We write the structural model in the general AB form as Ay t = AΠw t + Bɛ t, ɛ t N(, D), (5) where y t is a k 1 vector of endogenous variables, ɛ t is a k 1 vector of structural shocks, and w t = (1, y t 1,.., y t p) is a m 1 vector of the constant and the lagged variables, with m = kp+1 and p the number of lags. The matrix D is diagonal. Model (5) nests the A form for B = I k and the B form for A = I k, with I k the k k identity matrix. By contrast, when the model is specified in the AB form as in equation (5), 4

6 the corresponding A and B representations are Ãy t = ÃΠw t + ɛ t, ɛ t N(, D), (6) y t = Πw t + Bɛ t, ɛ t N(, D), (7) with à = B 1 A and B = A 1 B. The reduced form representation is y t = Πw t + u t, u t N(, Σ), (8) with u t = A 1 Bɛ t and Σ = A 1 BDB A 1. The k k matrices A, B and D include structural parameters, while Π and Σ represent reduced form parameters. We depart from the literature and parametrize the autoregressive component of the model as AΠ rather than in the more compact form C = AΠ in order to minimize the number of parameters for which the MCMC posterior sampler is required. 3 Parametrizing the model in Π makes it straightforward to apply the popular prior on the reduced form autoregressive coefficients by Litterman (1986) and Doan et al. (1984), because Π contains the parameters on which such popular prior was specified. When parametrizing the model as C = AΠ, this prior can be used only indirectly, as in Sims and Zha (1998) and Baumeister and Hamilton (15). To simplify the derivations of the posterior, we rewrite the model in more compact notation and vectorize Π, obtaining ỹ = (W I k )π + (I T A 1 B) ɛ, ɛ N (, (I T D) ). (9) 3 In the special case of the A form, it is more frequent to parametrize the autoregressive component of the model more compactly using C = AΠ, as in equation (1), rather than as AΠ. In fact, as shown by Sims and Zha (1998) and Baumeister and Hamilton (15), when the prior p(a) is independent across the equations of the model, posterior simulation for C and D can be done equation by equation from posterior distributions that take common forms, while the MCMC algorithm is only required for the elements in A. In the more general framework considered in this paper, the presence of B I k prevents from breaking the analysis equation by equation. Parametrizing the autoregressive component of the model as AΠ keeps the analysis tractable, because the posterior distribution p(π A, B, D, Y ) has a common form and can be drawn from using available random number generators, as discussed below. 5

7 The vector π, of dimension km 1, stacks the column vectors of Π, ỹ and ɛ are vectors of dimension kt 1 stacking the vectors y 1,..., y T and ɛ 1,.., ɛ T, while W = [w 1,.., w T ] is a matrix of dimension m T. With this parametrization, the joint prior distribution of the model is, without loss of generality, p(π, A, B, D) = p(π A, B, D) p(a, B, D). (1) As common in the literature, we restrict p(π A, B, D) to π N(µ π, V π ), (11) where µ π and V π can potentially be functions of A, B, D. By not imposing a Kronecker structure to the variance term, p(π A, B, D) allows for the setting of the hyperparameters µ π and V π to replicate the popular prior by Litterman (1986), treating the variance on own lags and lags on other variables differently. Contrary to p(π A, B, D), p(a, B, D) can fall within a wide range of prior distributions, granting the researcher flexibility on the prior information on structural parmeters. Given p(π, A, B, D), the joint posterior distribution p(π, A, B, D Y ) = p(π A, B, D, Y ) p(a, B, D Y ) (1) satisfies π A, B, D, Y N(µ π, V π ), (13) p(a, B, D Y ) p(a, B, D) det(a 1 B) T det(d) T [ Vπ = e 1 V 1 µ π = V π { µ π V π 1 det(vπ ) 1 det(v π ) 1 µ π +ỹ (I T (A B 1 D 1 B 1 A))ỹ+µ π V 1 π µ π }, (14) π + [ XX (A 1 BDB A 1 ) 1]] 1, (15) [ µ π + [ X (A 1 BDB A 1 ) 1] ] ỹ. (16) V 1 π 6

8 Accordingly, the analysis of the joint posterior distribution requires an MCMC algorithm only for the k + k elements in p(a, B, D Y ). In a representative AB model with 3 variables and 13 lags, this corresponds to 1 parameters out of a total of 141 parameters, well within the range in which the dynamic-strated metropolis-hastings sampler used performs efficiently. 4 The number of parameters requiring the MCMC sampler further decreases if one adopts the A form or the B form or if some elements of A or B are point identified, or if normalizations are introduced. 5 Conditioning on A, B and D, draws from π can be obtained with a standard random number generator. We refer the reader to appendix 6.1 for the derivations of the posterior distribution. We conclude this section by discussing the prior distribution p(a, B, D). p(a, B, D) reflects the identifying restrictions imposed on the structural parameters. Whether the researcher has sufficient information to specify p(a, B, D) depends on the economic question at hand. The shape of the prior on structural parameters can be suggested by economic theory. For example, if some elements of A bear the economic interpretation of price elasticities, external information on such elasticities can be used for the specification of the prior, as in Baumeister and Hamilton (15). A similar approach is also employed by Arias et al. (15), who use restrictions on the structural parameters in the monetary policy rule. However, as remarked by Kilian and Lütkepohl (17), it frequently happens that the researcher lacks information to specify a full prior distribution. This tends to be the case when identifying restrictions are imposed on the impact effect of a matrix, equation (7). For example, one might be willing to impose that an exogenous monetary expansion increases inflation, but is not in a position to specify the scale of such increase. The applications in this paper cover both of these scenarios. In Section 3 we use the 4 Waggoner et al. (16) apply their sampler to all the structural parameters of their model, showing that the sampler is successful also when exploring jointly more than 1 parameters. A back-of-the-envelope calculation shows that we could run our approach on a model with as many as 7 variables and still apply the posterior sampler on fewer parameters than in Waggoner et al. (16). How the approach developed in this paper performs in large scaled models is a question that we leave for future research. 5 In the AB form, one can add up to k normalizations. 7

9 the B form of the model and specify our prior under the hypothesis that the researcher has prior information on the sign of the impulse responses, but cannot use economic theory to guide his or her prior beliefs about the magnitude of the responses. Since this scenario arises frequently in applied research, we suggest a flexible approach that uses a training sample to estimate a plausible scale of the effects of the shocks, and construct truncated distributions that depend on two hyperparameters. Borrowing from the Minnesota prior, this approach combines the information from a training sample with a handful of hyperparameters to obtain Bayesian shrinkage while still granting the researcher with extensive flexibility for the specification of the prior. We refer to Section 3. for a detailed discussion and for an illustrative example. Then, in Section 4, we apply the AB model and use prior information from economic theory to specify the prior on the structural parameters. 3 An application to the New Keynesian model - the SVAR in B form In this section we apply the procedure discussed in Section to simulated data from the New Keynesian model. We discuss how we specify the prior distribution on the structural parameters and then compare the results using the orthogonal reduced form approach. In this section, we use the B form of the model. The AB form, instead, is used in Section 4 on real data. 3.1 The data generating process We use the benchmark specification of the linearized DSGE model by An and Schorfheide (7) as our data generating process. The model economy includes three expecta- 8

10 tional equations constraining three endogenous variables. The model is given by x t = E t x t+1 + g t E t g t+1 1 τ (r t E t π t+1 E t z t+1 ), π t = βe t π t+1 + κ(x t g t ), r t = ρ r r t 1 + (1 ρ r )η 1 π t + (1 ρ r )η (x t g t ) + ɛ rt, g t = ρ g g t 1 + ɛ gt, z t = ρ z z t 1 + ɛ zt. (17a) (17b) (17c) (17d) (17e) The variables of the model are the output gap (x t ), inflation (π t ), and the interest rate (r t ). The simulated data are driven by three uncorrelated structural shocks. These shocks are the interest rate shock ɛ rt N(, σ r), the government spending shock ɛ gt N(, σ g) and the technology shock ɛ zt N(, σ z). The fundamental parameters of the model are θ = (τ, r A, κ, ρ r, ρ g, ρ z, ψ 1, ψ, σ r, σ g, σ z ), with β = 1 1+r A /4.6 In the baseline specification we calibrate θ using the parameter values that An and Schorfheide (7) employ for their data generating process. These values are shown in Table 1. For completeness, for each parameter we also report the distribution that we employ in Setion 3.3 to assess the performance of our approach with alternative parametrizations of the model. These distributions coincide with the prior distributions that An and Schorfheide (7) use for the estimation of the model. Table 1: Calibration of the Data Generating Process Baseline Sensitivity analysis Baseline Sensitivity analysis mean std mean std τ Gamma.5 ρ z.65 Beta κ.15 Gamma..1 r A.4 Gamma.5.5 η Gamma σ r. InvGamma η 1 Gamma.5.5 1σ g.8 InvGamma ρ r.6 Beta.5. 1σ z.45 InvGamma ρ g.95 Beta To calibrate the model as in An and Schorfheide (7), we treat β as a function of the fundamental parameter r A, which determines the steady state interest rate, and parametrize the model using κ, which relates to fundamental parameters unreported in our application. 9

11 After calibrating the model, we use the solution method by Sims () to solve the model, and then the factorization by Fernandez-Villaverde et al. (7) to compute the associated VAR representation. For all calibrations used in our paper, the model has the following exact VAR(1) representation in the B form, y t = Π(θ)y t 1 + B(θ)ɛ t, (18) ɛ t N(, D(θ)), (19) with y t = (r t, x t, π t ) the 3 1 vector of endogenous variables and Π(θ), B(θ), D(θ) the VAR parameters For the baseline calibration used, the true model is r t x t π t }{{} y t r t = x t π t }{{}}{{}}{{} ɛ r t ɛ g t ɛ π t }{{} ɛ t Π(θ) N, y t 1 B(θ) }{{} D(θ) ɛ r t ɛ g t ɛ π t }{{} ɛ t On impact, a monetary shock of positive size increase the interest rate, output and inflation, a technology shock of positive sign increases the interest rate and decreases output and inflation, while a government spending shock only increases spending. The data generating process from the previous section is used to draw 8 observations, as in An and Schorfheide (7). () 1

12 3. Model, prior specification, and posterior sampler We use the generated data to estimate a SVAR model. For this section, we use the model in B form, after normalizing the variance of the structural shocks to the identity matrix. To ensure more generality in the application, we include a constant and 4 lags of the dependent variables rather than employing the information that the true model features no constant and only one lag. As a prior for π, the reduced form autoregressive parameters, we use the prior distribution from equation (11), setting µ π and V π as applied by Litterman (1986) for the Minnesota Prior. We set µ π to imply that each variable follows a white noise process, in accordance with the nontrending nature of the pseudo data generated. We then set V π as discussed in Canova (7), allowing each lagged coefficient to have different variance depending on whether it enters the equation of the variable itself or of another variable. Since this prior on π is well established in the literature, we refer the reader to Bańbura et al. (1), Canova (7), Koop et al. (1) and Kilian and Lütkepohl (17) for a discussion. It then remains to specify the prior distribution for B. We impose the sign restrictions that an expansionary technology shock increases the interest rate, output and inflation, and that a contractionary monetary shock increases the interest rate and decreases output and inflation. These sign restrictions, consistent with the data generating process, reflect the identifying restrictions of the model, but are not sufficient to specify a prior distribution for p(b). As discussed in Section, the approach developed in the paper offers great flexibility in the specification of p(b). In this application, we propose one possible specification of p(b) in order to implement the above sign restrictions. Call b ij the ij entry of matrix B in equation (7). b ij captures the effect of shock j on variable i. Define two hyperparameters ψ 1 and ψ, and call γ i a positive scalar that summarizes a realistic scale for variable i. Upon setting γ i, which we discuss below, p(b) can be constructed as p(b) = p(b ij ), with p(b ij ) as follows. If b ij is 11

13 restricted to be positive, depart from a normal distribution with mean equal to ψ 1 γ i and calibrate the variance such that the distribution truncated on the positive support has 95% prior mass in the space (, ψ γ i ). If b ij is restricted to be negative, depart from a normal distribution with mean equal to ψ 1 γ i and calibrate the variance such that the distribution truncated on the negative support has 95% prior mass in the space ( ψ γ i, ). If, instead, no sign restriction is imposed on b ij, set the mean of the normal distribution to and calibrate its variance such that 95% of the prior mass is in the space ( ψ γ i, ψ γ i ). Given γ i, the researcher can control the location and tightness of the priors using the hyperparameters ψ 1 and ψ. We compute γ i as follows. Since our application uses model (5) under the restrictions A = D = I, the covariance restrictions Σ = A 1 BDB A 1 simplify to Σ = BB. These can be shown to imply Σ.5 ii b ij Σ.5 ii. 7 We therefore set γ i = ˆΣ.5 ii, where ˆΣ is an estimate of Σ based on a training sample that employs % of the sample, as in Primiceri (5). Roughly speaking, this procedure extends to the structural parameters the procedure used within the Minnesota Prior for the variance of the reduced form parameters. With the Minnesota Prior, one selects a scale of each variable by computing the variance σ i of the residual on univariate AR processes on each variable, and then combines such scaling factor with a small number of hyperparameters to achieve Bayesian shrinkage. In our framework, the scaling factor is computed by exploiting the information on the covariance restrictions Σ = BB. An alternative approach is to set γ i = σ i. In the baseline analysis below, we set γ i = ˆΣ.5 ii, ψ 1 =.5 and ψ = 1.5 and consider alternative values in the prior sensitivity analysis. Alternatively, ψ 1 and ψ can be treated as random variables and studied using a hierarchical prior. We leave this extension to future research. An illustration of the calibration of p(b ij ) is shown in Figure 1, which displays the 7 When working on the A model with D = I, the covariance restrictions imply Σ.5 ii a ij Σ.5 ii with Σ I ii the ii entry of ΣI = Σ 1. When working with the AB model with D = I, the covariance restrictions imply Ω.5 ii b ij Ω.5 ii, with Ω = AΣA. See Piffer (16), footnote in Appendix D, for the derivations in the A and B model. 1

14 case for b 11. This parameter is sign restricted to be positive. Panel a shows a normal distribution centred at ψ 1 γ 1 (indicated with a square), with γ 1 =.17 (indicated with an x). Panel b then shows the distribution in panel a, truncated to be positive. The standard deviation of the distribution in panel a is calibrated such that 9% of the prior mass of the truncated distribution falls between zero and ψ γ 1 (the latter indicated with a diamond). Accordingly, ψ 1 and ψ control for the mode and the variance of the prior distribution p(b 11 ), given γ i. Higher values of ψ make the prior distribution more spread (panel c) while higher values of ψ 1 increase the prior mode (panel d). The prior distribution used in the analysis is then shown in Figure, together with the true value of B and the value of γ i. 8 We sample the posterior distribution from equation (14) using the dynamic striated Metropolis-Hastings sampler by Waggoner et al. (16). The sampler is particularly suitable in our application because it allows for the exploration of posterior distributions that potentially behave erratically and present several ridges. We outline the key ingredients of the algorithm in?? and discuss there how we calibrate its key parameters. Replication codes for Matlab and Fortran are made available on our websites. 3.3 Results The discussion of the results of this application are reported in three steps. First, we discuss the update in our approach and compare the posterior distribution to the true impulse responses. Second, we compare the result to what is instead obtained by applying the orthogonal reduced form approach to the same model and with the same identifying restrictions. Third, we run the analysis with alternative calibrations and extractions of the shocks to ensure that the results are not driven by an exceptional set of parameter values and shocks. Figures 3 and 4 show the update of the prior distribution on B and on the impulse 8 The true value of B reported in the figure is the ij entry of matrix B norm = B(θ)D(θ).5, to ensure consistency with the normalization used in the estimated model. 13

15 responses, respectively. The marginal posteriors of the entries of B tend to cover the true value, and in most cases display a clear shrinkage relative to the prior. The prior distribution on the impulse responses clearly display the sign restrictions on B, and have prior mass both for positive and for negative values of the responses from the second period onwards. As shown, the posterior distribution tends to cover the true impulse responses both on impact and at future horizons of the response. Overall, the update successfully uncovers the true structural dynamics associated with the identified shocks. Figures 5 and 6 then compare the results to the case in which the orthogonal reduced form approach is used. For this, we use the independent Normal-inverse Wishart prior and set the Normal prior on π exactly as with our approach, in order to ensure comparability. We consider two specifications of the inverse Wishart distribution for Σ, the first one to replicate the uninformative prior, the second one to replicate the commonly used specification by Kadiyala and Karlsson, These are widely used specifications of the prior and limit any subjectivity in the specification of the hyperparameters, making the comparison more objective. 9 Figures 5 and 6 show that, overall, all three approaches successfully deliver a posterior impulse response band that covers the true impulse responses. However, the approach proposed in this paper delivers posterior impulse response bands that are much tighter around the true impulse response. We interpret this result in light of the additional flexibility that our approach offers to the researcher. As discussed, for instance by Rossi et al. (1), the Wishart and inverse Wishart distributions have the undesirable property that, up to an approximation, a single parameter sets 9 We set the prior distribution for µ as in equation (11), with hyperparameters values as in our approach. For Σ we use the prior iw (S, ν), with ν = k and S =.5I k for the uninformative prior and with ν = k + 1 and S = diag((σ1,.., σk )) for the prior as in Kadiyala and Karlsson (1997). We simulate from the joint posterior distribution using a Gibbs sampler with, draws, burning in the first 1,. We then apply sign restrictions by randomly selecting a combination of the posterior draws for µ and Σ, drawing an orthogonal matrix, checking if the sign restrictions are satisfied, otherwise repeating draw of all three elements, until 1, successful draws were obtained. Drawing more than one orthogonal matrix per combination of reduced form parameters did not affect the results. 14

16 the tightness of the entire distribution. By contrast, specifying the prior distribution directly on B allows for extensive flexibility, which we make use of as discussed in Section 3.. We find that this additional flexibility can deliver an efficiency gain in the attempt to uncover the true dynamic impulse responses of the shocks. Table : Log-scores evaluated at the true IRF, differences from the orthogonal reduced form approach Difference from reduced form parametrization (uninformative) lags a) TFP shock b) Monetary shock e) TFP+monetary r t x t π t r t, x t, π t r t x t π t r t, x t, π t r t, x t, π t Difference from reduced form parametrization (Kadiyala and Karlsson, 1997) lags a) TFP shock b) Monetary shock e) TFP+monetary r t x t π t r t, x t, π t r t x t π t r t, x t, π t r t, x t, π t Notes: A negative difference indicates that, evaluated at the true impulse response, the posterior from the approach propose in the paper takes a higher value than the posterior obtained from the orthogonal reduced form approach using for Σ an uninformative prior (top panel) or the prior as in Kadiyala and Karlsson (1997) (lower panel). To further assess whether our approach recovers the true impulse responses more or less successfully than the orthogonal reduced form approach, we compute log scores at the true impulse response for the three cases considered, namely the direct structural approach in this paper, the orthogonal reduced form approach with an uninformative inverse Wishart prior, and the orthogonal reduced form approach with invere Wishart prior as in Kadiyala and Karlsson (1997). More specifically, we compute the value of the log of the posterior distribution and evaluate it in correspondence to the true 15

17 Table 3: Log-scores evaluated at the true IRF, share of Monte Carlo draws that feature a negative difference from the reduced-form-orthogonal parametrization Difference from reduced form parametrization (uninformative) lags a) TFP shock b) Monetary shock e) TFP+monetary r t x t π t r t, x t, π t r t x t π t r t, x t, π t r t, x t, π t Difference from reduced form parametrization (Kadiyala and Karlsson, 1997) lags a) TFP shock b) Monetary shock e) TFP+monetary r t x t π t r t, x t, π t r t x t π t r t, x t, π t r t, x t, π t Notes:. value. The differences, shown in Table, are computed such that a negative difference implies that the posterior distribution from our approach takes a higher value than the alternative approach indicated in each section of the table, when evaluated in correspondence to the true impulse response. The differences are computed for each shock and each variable in separation, for different lags, and for both shocks and all variables at all horizons considered in Figure 4. As clear from the table, the differences are largely negative, suggesting that, overall, the approach proposed in the paper is more successful in uncovering the true impulse response. Last, we assess whether the better performance of the approach delivered in this paper relative to the orthogonal reduced form approach is the outcome of a specific calibration used for the data generating process and/or a specific draw of the structural shocks or not. To do so, we replicate the log score analysis in a Monte Carlo experiment. Each time, we draw the parameters of the model from Section 3 from the prior distributions used in An and Schorfheide (7) and shown in Table 1. We 16

18 keep the parameter draws as long as they imply an exact VAR(1) representation of the model and satisfy the same sign patterns of the shocks as in the baseline calibration, otherwise we draw again. For each draw we randomly generate a dataset extracting new shocks, and derive the posterior impulse responses following our approach as well as both specifications of the prior on Σ under the orthogonal reduced form approach. We then compute the difference in log scores, following the discussion for Table. We repeat the analysis until successful draws are obtained. The results of the Monte Carlo exercise are shown in Table 3. The table shows the percentage of the total draws that are associated with a negative difference in the log score. As clear from the table, the differences are negative in the vast majority of Monte Carlo draws, further supporting the approach derived in the paper. When considering both shocks and all variables and horizons, the log differences were negative effectively for the entire set of Monte Carlo draws. 4 An application to fiscal shocks - the SVAR in AB form In this section we apply the procedure discussed in to real data. We present the model by Blanchard and Perotti () in AB form, discuss suitable prior distribution and compare the results from our model to those obtained in their original study. 4.1 The Blanchard & Perotti () Model Blanchard and Perotti () aim at characterising the dynamic effects of exogenous fiscal expansions, i.e. tax cuts and government spending increases, on US output. They achieve point identification by imposing numerical restrictions on fiscal rules. They find that on impact output responds stronger to spending than to tax shocks and that the response to spending shocks is longer lasting. 17

19 In a first step, they estimate a reduced-form VAR setting y t = [T t, G t, X t ], where T t are taxes, G t government spending, and X t GDP, all in quarterly, real, per capita logarithmic terms. The sample spans 195Q1 till 6Q4 and they use a lag length of 4 quarters. The total number of parameters to identify is 1 (9 in A and B, respectively, and 3 in D). The reduced-form covariance matrix, Σ, delivers 6 independent restrictions and they normalise the diagonal elements of A and B to unity, so they need 9 additional restrictions for point identification. Their system can be written in AB form as follows 1 : 1 a 1 1 b 1 c 1 c 1 V ar u taxes t u spending t u gdp t ɛ taxes t ɛ spending t ɛ gdp t 1 a = b 1 1 d 1 = d d 3 ɛ taxes t ɛ spending t ɛ gdp t (1) () The zeros in A imply that the only macro variable that taxes and spending react to contemporaneously is output. The zeros in B imply that it takes the government more than a quarter to learn about output shocks (last column) and that output reacts to tax and spending shocks only to the extent that these move actual taxes and spending, i.e. are not offset by automatic stabilizers (last row). In addition, a 1, the output elasticity of taxes, is calibrated to.8 based on extraneous data and b 1, the output elasticity of spending, is set to based on the argument that government 1 Note the relation between reduced form and structural shocks resulting from the AB specification Ay t = AΠw t + Bɛ t = A(Πw t + u t ) = AΠw t + Bɛ t Au t = Bɛ t, 18

20 consumption and spending do not have built-in stabilizers. a or b are restricted to alternately depending on whether one assumes that the tax or the spending decision occurs first. There is some existing literature criticising the numerical restrictions on fiscal rules (see e.g. Mertens and Ravn, 14 and Caldara and Kamps, 17) because varying 11 them leads to a broad range of fiscal multipliers. Especially the parameters a 1 and b 1 have been shown to severely influence the results. Therefore, relaxing the strict numerical restrictions and instead expressing priors directly on the relevant elasticities and impact effects of policy shocks to achieve set identification makes the analysis more tractable and encompasses the identification restrictions of a variety of studies. 4. Priors We avoid setting numerical restrictions and instead use independent prior distributions for the elements a 1, b 1, c 1, a, b, c, d 1, d, and d 3. As in the original study, the diagonal elements of A and B are normalized to 1 and the remaining elements in A, B and D are calibrated to. We make use of the previous literature to guide our choice of prior parameters and employ a sufficiently high variance to allow the data to update the prior. Table 4: Prior Densities Distribution Mode 95 % Probability Interval a 1 Truncated Normal.8 [; 7] a Normal [-.5;.5] b 1 Normal [-1; 1] b Normal [-.5;.5] c 1 Truncated Normal -1 [-; ] c Truncated Normal 1 [; ] d i Gamma τ i /(κ i + 1) - 11 We follow Blanchard and Perotti () and define the tax (spending) multiplier as the Dollar response of output to a tax (spending) shock of size 1 Dollar. 19

21 Table 4 shows the densities used for the free parameters. We follow the literature and specify the elements of D as inverse Gamma distributions: d i Γ 1 (κ i, τ i ), i = 1,, 3 (3) where κ i = 3 for all i to avoid infinite variances and τ i is set such that V ar(d i ) is equal to the variance of univariate AR(4) processes on a training sample up to 196Q4, as for the Minnesota prior. Truncated normal priors are used for the remaining elements where both mean and variance are informed by the existing literature on tax and spending multipliers. For a 1, an upper bound in the literature is 4. We employ a truncated normal where the mode is set to the calibration by Blanchard and Perotti (),.8, and variance such that 95% probability mass lie in the interval [; 7]. For b 1, based on the argument that spending does not have built-in stabilizers, we employ a relatively tight prior and use a normal distribution centered around and with 95% probability mass in the interval [-1; 1]. For a and b, given that the previous literature (see e.g. Caldara and Kamps, 17) found little interaction between taxes and spending, we employ untrancated normal distributions centered around with variances such that 95 % probability mass lie within the interval [-.5;.5]. For c 1 and c it is rarely found that these parameters exceed 1 in absolute value. Therefore, for c 1 we use a truncated normal with mode equal to -1 and variance such that 95 % of the probability mass lie in the interval [-; ]. For c we use a truncated normal with mode equal to 1 and variance such that 95% of the probability mass lie in the interval [; ]. 4.3 Results Figure 1 shows the Bayesian updating of the parameters in A, B and D. For both c 1 and c we find that the data moves the prior towards slightly higher absolute values. For a 1, the updating is strongest as the data strongly rejects low values for

22 the tax response to output such as the calibration of Blanchard and Perotti (), which is.8. Instead, we obtain a posterior median of about 8. For b 1, we obtain a posterior median of about -.5 meaning that the data are informative about the spending response to output and suggest a spending decrease following a positive output residual. Both a and b are not moved away from by much suggesting that tax and spending decisions are conducted independently. For the variance parameters, d i, although the prior is very flat, we obtain that their posterior densities are close to. Figure 13 shows the dynamic tax and spending multiplier for the original study by Blanchard and Perotti () and those obtained from our approach. Contrary to Caldara and Kamps (17) and Blanchard and Perotti () we find that tax decreases stimulate economic activity more than spending increases. We find the on-impact tax multiplier to be around 9 and it peaks at 13 after 4 quarters. The onimpact spending multiplier is around 1, consistent with Blanchard and Perotti () and Caldara and Kamps (17) while it peaks at 1. after 3 quarters. 5 Conclusions We show how to impose prior knowledge within structural VAR models both on the contemporaneous relations among variables and on the on impact effects of structural shocks at the same time. In so doing we offer applied researchers a flexible way to inform their estimation, whether they work in the so-called A model, the B model or the AB model. In a simulation exercise using the New Keynesian model we show that our approach recovers the true impulse response functions more tightly than the widely used orthogonal reduced form approach for set identification. In an application to fiscal shocks à la Blanchard and Perotti (), we show how tax and spending multipliers can be estimated without imposing questionable 1

23 numeric restrictions. We find that tax cuts stimulate the economy more than spending increases.

24 References Amisano, G. and Giannini, C. (1), Topics in structural VAR econometrics, Springer Science & Business Media. An, S. and Schorfheide, F. (7), Bayesian analysis of DSGE models, Econometric reviews 6(-4), Arias, J., Caldara, D. and Rubio-Ramírez, J. F. (15), The systematic component of monetary policy in svars: an agnostic identification procedure. Arias, J., Rubio-Ramirez, J. F. and Waggoner, D. F. (forthcoming), Inference based on SVARs identified with sign and zero restrictions: Theory and applications. Bańbura, M., Giannone, D. and Reichlin, L. (1), Large bayesian vector auto regressions, Journal of Applied Econometrics 5(1), Baumeister, C. and Hamilton, J. D. (15), Sign restrictions, structural vector autoregressions, and useful prior information, Econometrica 83(5), Blanchard, O. and Perotti, R. (), An empirical characterization of the dynamic effects of changes in government spending and taxes on output, the Quarterly Journal of economics 117(4), Bruns, M. and Piffer, M. (17), Sign restrictions in smooth transition VAR models. Caldara, D. and Kamps, C. (17), The analytics of SVARs: a unified framework to measure fiscal multipliers, The Review of Economic Studies 84(3), Canova, F. (7), Methods for applied macroeconomic research, Vol. 13, Princeton University Press. Dedola, L. and Neri, S. (7), What does a technology shock do? a VAR analysis with model-based sign restrictions, Journal of Monetary Economics 54(),

25 Doan, T., Litterman, R. and Sims, C. (1984), Forecasting and conditional projection using realistic prior distributions, Econometric reviews 3(1), 1 1. Fernandez-Villaverde, J., Rubio-Ramírez, J. F., Sargent, T. J. and Watson, M. W. (7), ABCs (and Ds) of understanding VARs, American Economic Review 97(3), Kadiyala, K. R. and Karlsson, S. (1997), Numerical methods for estimation and inference in bayesian VAR-models, Journal of Applied Econometrics pp Kilian, L. and Lütkepohl, H. (17), Structural vector autoregressive analysis, Cambridge University Press. Kociecki, A. (1), A prior for impulse responses in bayesian structural VAR models, Journal of Business & Economic Statistics 8(1), Kociecki, A., Rubaszek, M. and Ca Zorzi, M. (1), Bayesian analysis of recursive SVAR models with overidentifying restrictions. Koop, G., Korobilis, D. et al. (1), Bayesian multivariate time series methods for empirical macroeconomics, Foundations and Trends R in Econometrics 3(4), Litterman, R. B. (1986), Forecasting with bayesian vector autoregressions five years of experience, Journal of Business & Economic Statistics 4(1), Lütkepohl, H. (5), New introduction to multiple time series analysis, Springer Science & Business Media. Mertens, K. and Ravn, M. O. (14), A reconciliation of svar and narrative estimates of tax multipliers, Journal of Monetary Economics 68, S1 S19. Piffer, M. (16), Bayesian model comparison for sign restrictions in SVAR models. 4

26 Plagborg-Møller, M. et al. (16), Bayesian inference on structural impulse response functions, manuscript, Harvard University. Primiceri, G. E. (5), Time varying structural vector autoregressions and monetary policy, The Review of Economic Studies 7(3), Rossi, P. E., Allenby, G. M. and McCulloch, R. (1), Bayesian statistics and marketing, John Wiley & Sons. Sims, C. A. (), Solving linear rational expectations models, Computational economics (1), 1. Sims, C. A. and Zha, T. (1998), Bayesian methods for dynamic multivariate models, International Economic Review pp Sims, C. A. and Zha, T. (6), Were there regime switches in us monetary policy?, American Economic Review 96(1), Uhlig, H. (5), What are the effects of monetary policy on output? results from an agnostic identification procedure, Journal of Monetary Economics 5(), Waggoner, D. F., Wu, H. and Zha, T. (16), Striated metropolis hastings sampler for high-dimensional models, Journal of Econometrics 19(),

27 Figure 1: Illustrative example: p(b 11 ) Panel a) Panel b) i 1 i i Panel c) 1 Panel d) 8 1 =.5, = 1.5 (baseline) 1 =.5, = =.5, = 1.5 (baseline) 1 = 1, = Note: ADD. 6

28 Inflation Output gap Interest rate Figure : Prior distribution for B TFP shock Monetary shock 1 DGP i Note: ADD

29 Inflation Output gap Interest rate Figure 3: Prior and posterior distribution for B 8 6 TFP shock 8 6 Monetary shock 4 Prior Posterior Note: ADD

30 Inflation Inflation Output gap Output gap Interest rate Interest rate Figure 4: Prior and posterior distribution for the IRFs 4 TFP shock 4 Monetary shock True IRF Prior (68%) Posterior (68%) Note: ADD. 9

31 Inflation Inflation Output gap Output gap Interest rate Interest rate Figure 5: Compare to priors on reduced form (uninformative) 6 4 TFP shock Monetary shock True IRF Posterior, baseline (68%) Posterior, alternative (68%) Note: It shows our posterior, their posterior and the truth. They include the truth, but only because the scale is huge, cannot even see our posterior. Their prior is too wide. 3

32 Inflation Inflation Output gap Output gap Interest rate Interest rate Figure 6: Compare to priors on reduced form (Kadiyala and Karlsson, 1997) 6 4 TFP shock 6 4 Monetary shock True IRF Posterior, baseline (68%) Posterior, alternative (68%) Note: It shows our prior, their prior and their posterior: they have tighter prior, but posterior underestimates the truth. 31

33 6 Appendix 6.1 Derivation of the posterior distribution Depart from the model in equation (5), which we rewrite here as y t = Πw t + A 1 Bɛ t, (4) ɛ t N(, D). (5) The vectors y t and ɛ t are of dimension k 1, the vector w t is of dimension m 1, the matrix Π is of dimension k m, and the matrices A, B, D are of dimension k k. Rewrite equation (4) in compact form as Y = ΠW + A 1 BE, (6) with Y = [y 1,..., y t,..., y T ] of dimensions k T, W = [w 1,..., w t,..., w T ] of dimensions m T and E = [ɛ 1,..., ɛ t,..., ɛ T ] of dimensions k T. The compact form stacks the observations of each variables next to each other. Then, make use of the formula vec(ā B C) = ( C Ā) vec( B) (see Lütkepohl, 5, mathematical appendix) and rewrite (6) as ỹ = (W I k )π + (I T A 1 B) ɛ, (7) ( ) ɛ N, (I T D) (8) with ỹ = vec(y ) and ɛ = vec(e) of dimension kt 1. The vector µ = vec(π) stacks each column of Π vertically and is of dimension mk 1. Last, rewrite equation (7) 3

34 in reduced form, obtaining ỹ = (W I k )π + ũ, (9) ( ) ũ N, (I T A 1 BDB A 1 ). (3) The covariance matrix of ũ is obtained by simplifying the expression (I T A 1 B)(I T D)(I T A 1 B), given.... Define Z = (W I k ). From model (9)-(3), the likelihood function can be written as p(ỹ π, A, B, D) = (π) kt IT A 1 BDB A 1 1 e 1 (ỹ Zπ) (I T A 1 BDB A 1 ) 1 (ỹ Zπ), which simplifies to (31) p(ỹ π, A, B, D) = (π) kt A 1 B T D T e 1 (ỹ Zπ) (I T A B 1 D 1 B 1 A)(ỹ Zπ). (3) since I T A 1 BDB A 1 = A 1 B T D T and (I T A 1 BDB A 1 ) 1 = (I T A B 1 D 1 B 1 A). Given the prior distribution discussed in Section, the joint posterior distribution is p(π, A, B, D ỹ) =p(π, A, B, D) with p(ỹ) = π A p(ỹ π, A, B, D), p(ỹ) =(π) k Vπ 1 e 1 (π µ π ) V 1 π (π µ π ) p(a, B, D)p(ỹ) 1... (33) B (π) kt A 1 B T D T e 1 (ỹ Zπ) (I T A B 1 D 1 B 1 A) 1 (ỹ Zπ), D p(ỹ π, A, B, D)p(π, A, B, D)dπdAdBdD. We aim to rewrite the joint posterior distribution as p(π, A, B, D ỹ) = p(π A, B, D, ỹ) p(a, B, D ỹ), (34) 33

35 and to exploit analytical results for p(π A, B, D, ỹ). To do so, rewrite first the joint posterior distribution as p(π, A, B, D ỹ) = c e 1 [ (π µ π ) V 1 π (π µ π )+(ỹ Zπ) Ω(A,B,D) 1 (y Zπ) ], (35) with c a term that is constant in π. Factorize the relevant terms in the exponent of the above expression as (δ µ δ ) V 1 δ (δ µ δ ) + (y Zδ) Ω(B) 1 (y Zδ) = (36) (δ µ δ ) Ṽ 1 δ (δ µ δ ) + y Ω(B) 1 y µ δṽ 1 δ µ δ + µ δ V 1 δ µ δ, (37) with Ṽ δ = [V 1 δ + Z Ω(B, γ, c) 1 Z] 1, (38) µ δ = Ṽδ [V 1 δ µ δ + Z Ω(B, γ, c) 1 y]. (39) Hence, the joint posterior distribution can be equivalently expressed as p(δ, B Y ) =(π) n Ṽ δ 1 e 1 (δ µ 1 δ ) Ṽδ (δ µ δ)... p(b, γ, c) Ṽδ 1 Ω(B, γ, c) 1 e 1 {y Ω(B,γ,c) 1 y µ Ṽ 1 δ µ}... (4) V δ 1 p(y ) 1 (π) T n e µ δ V 1 δ µ δ. It follows that δ B, γ, c, Y N( µ δ, Ṽ 1 δ ), (41) while the kernel of p(b, γ, c Y ) satisfies 34